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Data Science and Predictive Analytics (UMich HS650)

Dimensionality Reduction

Now that we have most of the fundamentals covered in the previous chapters, we can delve into the first data analytic method, dimension reduction, which reduces the number of features when dealing with a very large number of variables. Dimension reduction can help us extract a set of “uncorrelated” principal variables and reduce the complexity of the data. We are not simply picking some of the original variables. Rather, we are constructing new “uncorrelated” variables as functions of the old features.

Dimensionality reduction techniques enable exploratory data analyses by reducing the complexity of the dataset, still approximately preserving important properties, such as retaining the distances between cases or subjects. If we are able to reduce the complexity down to a few dimensions, we can then plot the data and untangle its intrinsic characteristics.

All dimensionality reduction techniques carry some resemblance to variable selection, which we will see in Chapter 16. The continuous process of dimensionality reduction yields lower variability compared to the discrete feature selection process, e.g., see Wolfgang M. Hartmann’s paper.

We will (1) start with a synthetic example demonstrating the reduction of a 2D data into 1D, (2) explain the notion of rotation matrices, (3) show examples of principal component analysis (PCA), singular value decomposition (SVD), independent component analysis (ICA), factor analysis (FA), and t-distributed Stochastic Neighbor Embedding (t-SNE), and (4) present a Parkinson’s disease case-study at the end.

1 Example: Reducing 2D to 1D

We consider an example looking at twin heights. Suppose we simulate 1,000 2D points that representing normalized individual heights, i.e., number of standard deviations from the mean height. Each 2D point represents a pair of twins. We will simulate this scenario using Bivariate Normal Distribution.

library(MASS)

set.seed(1234)
n <- 1000
y=t(mvrnorm(n, c(0, 0), matrix(c(1, 0.95, 0.95, 1), 2, 2)))

\(Twin1_{Height}\)	\(Twin2_{Height}\)
\(y[1,1]\)	\(y[1,2]\)
\(y[2,1]\)	\(y[2,2]\)
\(y[3,1]\)	\(y[3,2]\)
\(\cdots\)	\(\cdots\)
\(y[500,1]\)	\(y[500,2]\)

\[ y^T_{2\times500} = \begin{bmatrix} y[1, ]=Twin1_{Height} \\ y[2, ]=Twin2_{Height} \end{bmatrix} \sim BVN \left ( \mu= \begin{bmatrix} Twin1_{Height} \\ Twin2_{Height} \end{bmatrix} , \Sigma=\begin{bmatrix} 1 & 0.95 \\ 0.95 & 1 \end{bmatrix} \right ) .\]

plot(y[1, ], y[2, ], xlab="Twin 1 (standardized height)", 
     ylab="Twin 2 (standardized height)", xlim=c(-3, 3), ylim=c(-3, 3))
points(y[1, 1:2], y[2, 1:2], col=2, pch=16)

These data may represent a fraction of the information included in a high-throughput neuroimaging genetics study of twins. You can see one example of such pediatric study here.

Tracking the distances between any two samples can be accomplished using the dist function. stats::dist() is a function computing the distance matrix (according to a user-specified distance metric, default=Euclidean) representing the distances between the rows of the data matrix. For example, here is the distance between the two RED points in the figure above:

d=dist(t(y))
as.matrix(d)[1, 2]

## [1] 2.100187

To reduce the 2D data to a simpler 1D plot we can transform the data to a 1D matrix (vector) preserving (approximately) the distances between the 2D points.

The 2D plot shows the Euclidean distance between the pair of RED points, the length of this line is the distance between the 2 points. In 2D, these lines tend to go along the direction of the diagonal. If we rotate the plot so that the diagonal aligned with the x-axis we get the following raw and tranformed plots:

z1 = (y[1, ]+y[2, ])/2 # the sum (actually average)
z2 = (y[1, ]-y[2, ])   # the difference

z = rbind( z1, z2) # matrix z has the same dimension as y

thelim <- c(-3, 3)
# par(mar=c(1, 2))
# par(mfrow=c(2,1))
plot(y[1, ], y[2, ], xlab="Twin 1 (standardized height)", 
     ylab="Twin 2 (standardized height)", 
     xlim=thelim, ylim=thelim)
points(y[1, 1:2], y[2, 1:2], col=2, pch=16)

Raw (above) and transformed (below) Twin heights scatterplots.

plot(z[1, ], z[2, ], xlim=thelim, ylim=thelim, xlab="Average height", ylab="Difference in height")
points(z[1, 1:2], z[2, 1:2], col=2, pch=16)

Raw (above) and transformed (below) Twin heights scatterplots.

par(mfrow=c(1,1))

Of course, matrix linear algebra notation can be used to represent this affine transformation of the data. Here we can see that to get the result z, we multiplied y by the matrix \(A\):

\[ A = \begin{pmatrix} 1/2&1/2\\ 1&-1\\ \end{pmatrix} \implies z = A \times y .\]

We can invert this transform by multiplying the result by the inverse (rotation matrix?) \(A^{-1}\) as follows:

\[ A^{-1} = \begin{pmatrix} 1&1/2\\ 1&-1/2\\ \end{pmatrix} \implies y = A^{-1} \times z \]

You can try this in R:

A <- matrix(c(1/2, 1, 1/2, -1), nrow=2, ncol=2); A   # define a matrix

##      [,1] [,2]
## [1,]  0.5  0.5
## [2,]  1.0 -1.0

A_inv <- solve(A); A_inv  # inverse

##      [,1] [,2]
## [1,]    1  0.5
## [2,]    1 -0.5

A %*% A_inv # Verify result

##      [,1] [,2]
## [1,]    1    0
## [2,]    0    1

Note that this matrix transformation did not preserve distances, i.e., it’s not a simple rotation in 2D:

d=dist(t(y)); as.matrix(d)[1, 2]     # distance between first two points in Y

## [1] 2.100187

d1=dist(t(z)); as.matrix(d1)[1, 2]   # distance between first two points in Z=A*Y

## [1] 1.541323

2 Matrix Rotations

One important question is how to identify transformations that preserve distances. In mathematics, transformations between metric spaces that are distance-preserving are called isometries (or congruences or congruent transformations).

First, let’s test the MA transformation we used above: \[A=\frac{Y_1+Y_2}{2} \\ M=Y_1 - Y_2.\]

MA <- matrix(c(1/2, 1, 1/2, -1), 2, 2)

MA_z <- MA%*%y
d <- dist(t(y))
d_MA <- dist(t(MA_z))

plot(as.numeric(d), as.numeric(d_MA)) 
abline(0, 1, col=2)

Observe that this MA transformation is not an isometry - the distances are not preserved. Here is one example with \(v1=\begin{bmatrix} v1_x=0 \\ v1_y=1 \end{bmatrix}\), \(v2=\begin{bmatrix} v2_x=1 \\ v2_y=0 \end{bmatrix}\), which are distance \(\sqrt{2}\) apart in their native space, but separated further by the transformation \(MA\), \(d(MA(v1),MA(v2))=2\).

MA; t(MA); solve(MA); t(MA) - solve(MA)

##      [,1] [,2]
## [1,]  0.5  0.5
## [2,]  1.0 -1.0

##      [,1] [,2]
## [1,]  0.5    1
## [2,]  0.5   -1

##      [,1] [,2]
## [1,]    1  0.5
## [2,]    1 -0.5

##      [,1] [,2]
## [1,] -0.5  0.5
## [2,] -0.5 -0.5

v1 <- c(0,1); v2 <- c(1,0); rbind(v1,v2)

##    [,1] [,2]
## v1    0    1
## v2    1    0

euc.dist <- function(x1, x2) sqrt(sum((x1 - x2) ^ 2))
euc.dist(v1,v2)

## [1] 1.414214

v1_t <- MA %*% v1; v2_t <- MA %*% v2
euc.dist(v1_t,v2_t)

## [1] 2

More generally, if \[ \begin{pmatrix} Y_1\\ Y_2\\ \end{pmatrix} \sim BVN \left( \begin{pmatrix} \mu_1\\ \mu_2\\ \end{pmatrix}, \begin{pmatrix} \sigma_1^2&\sigma_{12}\\ \sigma_{12}&\sigma_2^2\\ \end{pmatrix} \right)\] Then, \[ Z = AY + \eta \sim BVN(\eta + A\mu,A\Sigma A^{T}).\] Where BVN denotes bivariate normal distribution, \(A = \begin{pmatrix}a&b\\c&d\\ \end{pmatrix}\), \(Y=(Y_1,Y_2)^T\), \(\mu = (\mu_1,\mu_2)\), \(\Sigma = \begin{pmatrix} \sigma_1^2&\sigma_{12}\\ \sigma_{12}&\sigma_2^2\\ \end{pmatrix}\).

You can verify this by using the change of variable theorem. Thus, affine transformations preserve bivariate normality. However, in general, the linear transformation (\(A\)) is not guaranteed to be an isometry.

The question now is under what additional conditions for the transformation matrix \(A\), can we guarantee an isometry.

Notice that, \[ d^2(P_i,P_j) =\sum_{k=1}^{n} (P_{jk}-P_{ik})^2 = ||P||^2 = P^TP,\]

where \(P = (P_{j,1}-P_{i,1},...,P_{j,n}-P_{i,n})^T\), \(P_i,P_j \in R^n\) are any two points in \(n\) dimensions.

Thus, the only requirement we need for \(A\) to be an isometry is \((AY)^T(AY)=Y^TY\), i.e., \(A^TA=I\), which implies that \(A\) is an orthogonal (rotational) matrix.

Let’s use a two dimension orthogonal matrix to illustrate this.

Set \(A = \frac{1}{\sqrt{2}} \begin{pmatrix}1&1\\1&-1\\ \end{pmatrix}\). It’s easy to verify that \(A\) is an orthogonal (2D rotation) matrix.

The simplest way to test the isometry is to perform the linear transformation directly as follow.

A <- 1/sqrt(2)*matrix(c(1, 1, 1, -1), 2, 2)
z <- A%*%y
d <- dist(t(y))
d2 <- dist(t(z))

plot(as.numeric(d), as.numeric(d2)) 
abline(0, 1, col=2)

We can observe that the distances between points computed from original data and the transformed data are the same. Thus, the transformation \(A\) is called a rotation (isometry) of \(y\).

The alternative method is to simulate from joint distribution of \(Z = (Z_1,Z_2)^T\).

As we have mentioned above: \(Z = AY + \eta \sim BVN(\eta + A\mu,A\Sigma A^{T})\).

where \(\eta = (0,0)^T\), \(\Sigma = \begin{pmatrix} 1&0.95\\0.95&1\\ \end{pmatrix}\), \(A = \frac{1}{\sqrt{2}} \begin{pmatrix}1&1\\1&-1\\ \end{pmatrix}\).

We can compute \(A\Sigma A^{T}\) by hand or using matrix multiplication in R:

sig <- matrix(c(1,0.95,0.95,1),nrow=2)
A%*%sig%*%t(A)

##      [,1] [,2]
## [1,] 1.95 0.00
## [2,] 0.00 0.05

\(A\Sigma A^{T}\) represents the variance-covariance matrix, \(cov(z_1,z_2)\). We can simulate \(z_1\), \(z_2\) independently from \(z_1\sim N(0,1.95)\) and \(z_2 \sim N(0,0.05)\) (Note: independence and uncorrelation are equivalent for bivariate normal distribution).

set.seed(2017)
zz1 = rnorm(1000,0,sd = sqrt(1.95))
zz2 = rnorm(1000,0,sd = sqrt(0.05))
zz = rbind(zz1,zz2)
d3 = dist(t(zz))
qqplot(d,d3)
abline(a = 0,b=1,col=2)

Again, we see that the distances between pairs of points computed from original data and the simulated data coincide.

thelim <- c(-3, 3)
#par(mfrow=c(2,1))
plot(y[1, ], y[2, ], xlab="Twin 1 (standardized height)", 
     ylab="Twin 2 (standardized height)", 
     xlim=thelim, ylim=thelim)
points(y[1, 1:2], y[2, 1:2], col=2, pch=16)

Twin height scatterplot (upper) and after rotation (lower).

plot(z[1, ], z[2, ], xlim=thelim, ylim=thelim, xlab="Average height", ylab="Difference in height")
points(z[1, 1:2], z[2, 1:2], col=2, pch=16)

Twin height scatterplot (upper) and after rotation (lower).

par(mfrow=c(1,1))

We applied this transformation and observed that the distances between points were unchanged after the rotation. This rotation achieves the goals of:

• preserving the distances between points, and
• reducing the dimensionality of the data (see plot reducing 2D to 1D).

Removing the second dimension and recomputing the distances, we get:

d4 = dist(z[1, ]) ##distance computed using just first dimension
plot(as.numeric(d), as.numeric(d4)) 
abline(0, 1)

Distance computed with just one dimension after rotation versus actual distance.

The 1D distance provides a very good approximation to the actual 2D distance. This first dimension of the transformed data is called the first principal component. In general, this idea motivates the use of principal component analysis (PCA) and the singular value decomposition (SVD) to achieve dimension reduction.

3 Notation

In the notation above, the rows represent variables and columns represent cases. In general, rows represent cases and columns represent variables. Hence, in our example shown here, \(Y\) would be transposed to be an \(N \times 2\) matrix. This is the most common way to represent the data: individuals in the rows, features are columns. In genomics, it is more common to represent subjects/SNPs/genes in the columns. For example, genes are rows and samples are columns. The sample covariance matrix usually denoted with \(\mathbf{X}^\top\mathbf{X}\) and has cells representing covariance between two units. Yet for this to be the case, we need the rows of \(\mathbf{X}\) to represents units. Here, we have to compute, \(\mathbf{Y}\mathbf{Y}^\top\) instead following the rescaling.

4 Summary (PCA vs. ICA vs. FA)

Principle Component Analysis (PCA), Independent Component Analysis (ICA), and Factor Analysis (FA) are similar strategies, seeking to identify a new basis (vectors representing the principal directions) that the data is projected against to maximize certain (specific to each technique) objective function. These basis functions, or vectors, are just linear combinations of the original features in the data/signal.

The singular value decomposition (SVD), discussed later in this chapter, provides a specific matrix factorization algorithm that can be employed in various techniques to decompose a data matrix \(X_{m\times n}\) as \({U\Sigma V^{T}}\), where \({U}\) is an \(m \times m\) real or complex unitary matrix (\({U^TU=UU^T=I}\), i.e., \(|\det(U)|=1\)), \({\Sigma }\) is a \(m\times n\) rectangular diagonal matrix of singular values, representing non-negative values on the diagonal, and \({V}\) is an \(n\times n\) unitary matrix.

Method	Assumptions	Cost Function Optimization	Applications
PCA	Gaussian signals, linear bivariate relations	Aims to explain the variance in the original signal. Minimizes the covariance of the data and yields high-energy orthogonal vectors in terms of the signal variance. PCA looks for an orthogonal linear transformation that maximizes the variance of the variables	Relies on \(1^{st}\) and \(2^{nd}\) moments of the measured data, which makes it useful when data features are close to Gaussian
ICA	No Gaussian signal assumptions	Minimizes higher-order statistics (e.g., \(3^{rd}\) and \(4^{th}\) order skewness and kurtosis), effectively minimizing the mutual information of the transformed output. ICA seeks a linear transformation where the basis vectors are statistically independent, but neither Gaussian, orthogonal or ranked in order	Applicable for non-Gaussian, very noisy, or mixture processes composed of simultaneous input from multiple sources
FA	Approximately Gaussian data	Objective function relies on second order moments to compute likelihood ratios. FA factors are linear combinations that maximize the shared portion of the variance underlying latent variables, which may use a variety of optimization strategies (e.g., maximum likelihood)	PCA-generalization used to test a theoretical model of latent factors causing the observed features

5 Principal Component Analysis (PCA)

PCA (principal component analysis) is a mathematical procedure that transforms a number of possibly correlated variables into a smaller number of uncorrelated variables through a process known as orthogonal transformation.

Principal Components

Let’s consider the simplest situation where we have n observations \(\{p_1, p_2, ..., p_n\}\) with 2 features \(p_i=(x_i, y_i)\). When we draw them on a plot, we use x-axis and y-axis for positioning. However, we can make our own coordinate system by principal components.

ex<-data.frame(x=c(1, 3, 5, 6, 10, 16, 50), y=c(4, 6, 5, 7, 10, 13, 12))
reg1<-lm(y~x, data=ex)
plot(ex)
abline(reg1, col='red', lwd=4)
text(40, 10.5, "pc1")
segments(10.5, 11, 15, 7, lwd=4)
text(11, 7, "pc2")

Illustrated on the graph, the first PC, \(pc_1\) is a minimum distance fit in the feature space. The second PC is a minimum distance fit to a line perpendicular to the first PC. Similarly, the third PC would be a minimum distance fit to all previous PCs. In our case of a 2D space, two PC is the most we can have. In higher dimensional spaces, we need to consider how many PCs do we need to make the best performance.

In general, the formula for the first PC is \(pc_1=a_1^TX=\sum_{i=1}^N a_{i, 1}X_i\) where \(X_i\) is a \(n\times 1\) vector representing a column of the matrix \(X\) (representing a total of n observations and N features). The weights \(a_1=\{a_{1, 1}, a_{2, 1}, ..., a_{N, 1}\}\) are chosen to maximize the variance of \(pc_1\). According to this rule, the \(k^{th}\) PC is \(pc_k=a_k^TX=\sum_{i=1}^N a_{i, k}X_i\). \(a_k=\{a_{1, k}, a_{2, k}, ..., a_{N, k}\}\) has to be constrained by more conditions:

1. Variance of \(pc_k\) is maximized
2. \(Cov(pc_k, pc_l)=0\), \(\forall 1\leq l<k\)
3. \(a_k^Ta_k=1\) (the weights vectors are unitary)

Let’s figure out how to find \(a_1\). First we need to express the variance of our first principal component using the variance covariance matrix of \(X\): \[Var(pc_1)=E(pc_1^2)-(E(pc_1))^2=\] \[\sum_{i, j=1}^N a_{i, 1} a_{j, 1} E(x_i x_j)-\sum_{i, j=1}^N a_{i, 1} a_{j, 1} E(x_i)E(x_j)=\] \[\sum_{i, j=1}^N a_{i, 1} a_{j, 1} S_{i, j}.\]

Where \(S_{i, j}=E(x_i x_j)-E(x_i)E(x_j)\).

This implies \(Var(pc_1)=a_1^TS a_1\) where \(S=S_{i, j}\) is the covariance matrix of \(X=\{X_1, ..., X_N\}\). Since \(a_1\) maximized \(Var(pc_1)\) and the constrain \(a_1^T a_1=1\) holds, we can rewrite \(a_1\) as: \[a_1=max_{a_1}(a_1^TS a_1-\lambda (a_1^T a_1-1))\] Where the second part after the minus sign should be 0. To maximize this quadratic expression, we can take the derivative of this expression w.r.t. \(a_1\) and set it to 0. This yields \((S-\lambda I_N)a_1=0\).

In Chapter 4 we showed that \(a_1\) will correspond to the largest eigenvalue of \(S\), the variance covariance matrix of \(X\). Hence, \(pc_1\) retains the largest amount of variation in the sample. Likewise, \(a_k\) is the \(k^{th}\) largest eigenvalue of \(S\).

PCA requires the mean for each column in the data matrix to be zero. That is, the sample mean of each column is shifted to zero.

Let’s use a subset (N=33) of Parkinson’s Progression Markers Initiative (PPMI) data to demonstrate the relationship between \(S\) and PC loadings. First, we need to import the dataset into R and delete the patient ID column.

library(rvest)
wiki_url <- read_html("http://wiki.socr.umich.edu/index.php/SMHS_PCA_ICA_FA")
html_nodes(wiki_url, "#content")

## {xml_nodeset (1)}
## [1] <div id="content" class="mw-body-primary" role="main">\n\t<a id="top ...

pd.sub <- html_table(html_nodes(wiki_url, "table")[[1]])
summary(pd.sub)

##    Patient_ID   Top_of_SN_Voxel_Intensity_Ratio
##  Min.   :3001   Min.   :1.058                  
##  1st Qu.:3012   1st Qu.:1.334                  
##  Median :3029   Median :1.485                  
##  Mean   :3204   Mean   :1.532                  
##  3rd Qu.:3314   3rd Qu.:1.755                  
##  Max.   :3808   Max.   :2.149                  
##  Side_of_SN_Voxel_Intensity_Ratio    Part_IA         Part_IB      
##  Min.   :0.9306                   Min.   :0.000   Min.   : 0.000  
##  1st Qu.:0.9958                   1st Qu.:0.000   1st Qu.: 2.000  
##  Median :1.1110                   Median :1.000   Median : 5.000  
##  Mean   :1.1065                   Mean   :1.242   Mean   : 4.909  
##  3rd Qu.:1.1978                   3rd Qu.:2.000   3rd Qu.: 7.000  
##  Max.   :1.3811                   Max.   :6.000   Max.   :13.000  
##     Part_II          Part_III    
##  Min.   : 0.000   Min.   : 0.00  
##  1st Qu.: 0.000   1st Qu.: 2.00  
##  Median : 2.000   Median :12.00  
##  Mean   : 4.091   Mean   :13.39  
##  3rd Qu.: 6.000   3rd Qu.:20.00  
##  Max.   :17.000   Max.   :36.00

pd.sub<-pd.sub[, -1]

Then, we need to center the pdsub by subtracting the average of all column means from each element in the column. Next, we cast pd.sub as a matrix and compute its variance covariance matrix, \(S\). Finally, we can calculate the corresponding eigenvalues and eigenvectors of \(S\).

mu<-apply(pd.sub, 2, mean)
mean(mu)

## [1] 4.379068

pd.center<-as.matrix(pd.sub)-mean(mu)
S<-cov(pd.center)
eigen(S)

## eigen() decomposition
## $values
## [1] 1.315073e+02 1.178340e+01 6.096920e+00 1.424351e+00 6.094592e-02
## [6] 8.035403e-03
## 
## $vectors
##              [,1]          [,2]         [,3]        [,4]        [,5]
## [1,] -0.007460885 -0.0182022093  0.016893318  0.02071859  0.97198980
## [2,] -0.005800877  0.0006155246  0.004186177  0.01552971  0.23234862
## [3,]  0.080839361 -0.0600389904 -0.027351225  0.99421646 -0.02352324
## [4,]  0.229718933 -0.2817718053 -0.929463536 -0.06088782  0.01466136
## [5,]  0.282109618 -0.8926329596  0.344508308 -0.06772403 -0.01764367
## [6,]  0.927911126  0.3462292153  0.127908417 -0.05068855  0.01305167
##              [,6]
## [1,] -0.232667561
## [2,]  0.972482080
## [3,] -0.009618592
## [4,]  0.003019008
## [5,]  0.006061772
## [6,]  0.002456374

The next step would be calculating the PCs using the prcomp() function in R. Note that we will use the raw (uncentered) version of the data and have to specify the center=TRUE option to ensure the column means are trivial. We can save the model information into pca1 where pca1$rotation provides the loadings for each PC.

pca1<-prcomp(as.matrix(pd.sub), center = T)
summary(pca1)

## Importance of components:
##                            PC1    PC2     PC3     PC4    PC5     PC6
## Standard deviation     11.4677 3.4327 2.46919 1.19346 0.2469 0.08964
## Proportion of Variance  0.8716 0.0781 0.04041 0.00944 0.0004 0.00005
## Cumulative Proportion   0.8716 0.9497 0.99010 0.99954 1.0000 1.00000

pca1$rotation

##                                           PC1           PC2          PC3
## Top_of_SN_Voxel_Intensity_Ratio   0.007460885 -0.0182022093  0.016893318
## Side_of_SN_Voxel_Intensity_Ratio  0.005800877  0.0006155246  0.004186177
## Part_IA                          -0.080839361 -0.0600389904 -0.027351225
## Part_IB                          -0.229718933 -0.2817718053 -0.929463536
## Part_II                          -0.282109618 -0.8926329596  0.344508308
## Part_III                         -0.927911126  0.3462292153  0.127908417
##                                          PC4         PC5          PC6
## Top_of_SN_Voxel_Intensity_Ratio   0.02071859 -0.97198980 -0.232667561
## Side_of_SN_Voxel_Intensity_Ratio  0.01552971 -0.23234862  0.972482080
## Part_IA                           0.99421646  0.02352324 -0.009618592
## Part_IB                          -0.06088782 -0.01466136  0.003019008
## Part_II                          -0.06772403  0.01764367  0.006061772
## Part_III                         -0.05068855 -0.01305167  0.002456374

We notice that the loadings are just the eigenvectors times -1. These loadings represent vectors in 6D space (we have 6 columns in the original data). The scale factor -1 just represents the opposite direction of the eigenvector. We can also load the factoextra package an compute the eigenvalues of each PC.

# install.packages("factoextra")
library("factoextra")
eigen<-get_eigenvalue(pca1)
eigen

##         eigenvalue variance.percent cumulative.variance.percent
## Dim.1 1.315073e+02     87.159638589                    87.15964
## Dim.2 1.178340e+01      7.809737384                    94.96938
## Dim.3 6.096920e+00      4.040881920                    99.01026
## Dim.4 1.424351e+00      0.944023059                    99.95428
## Dim.5 6.094592e-02      0.040393390                    99.99467
## Dim.6 8.035403e-03      0.005325659                   100.00000

The eigenvalues correspond to the amount of the variation explained by each principal component (PC), which is the same as the eigenvalues of the \(S\) matrix.

To see a detailed information about the variances that each PC explain we utilize the plot() function to visualize the PC loadings.

plot(pca1)

library(graphics)
biplot(pca1, choices = 1:2, scale = 1, pc.biplot = F)

library("factoextra")
# Data for the supplementary qualitative variables
qualit_vars <- as.factor(pd.sub$Part_IA)
head(qualit_vars)

## [1] 0 3 1 0 1 1
## Levels: 0 1 2 3 4 6

# for plots of individuals
# fviz_pca_ind(pca1, habillage = qualit_vars, addEllipses = TRUE, ellipse.level = 0.68) +
#  theme_minimal()
# for Biplot of individuals and variables 
fviz_pca_biplot(pca1, axes = c(1, 2), geom = c("point", "text"),
  col.ind = "black", col.var = "steelblue", label = "all",
  invisible = "none", repel = T, habillage = qualit_vars, 
  palette = NULL, addEllipses = TRUE, title = "PCA - Biplot")

The histogram plot has a clear “elbow” point at the second PC, suggesting that the first two PCs explain about 95% of the variation in the original dataset. Thus, we say we can use the first 2 PCs to represent the data. In this case, the dimension of the data is substantially reduced.

Here, biplot uses PC1 and PC2 as the axes and red vectors to represent the direction of variables after adding loadings as weights. It help us to visualize how the loadings are used to rearrange the structure of the data.

Next, let’s try to obtain a bootstrap test for the confidence interval of the explained variance.

set.seed(12)
num_boot = 1000
bootstrap_it = function(i) {
  data_resample = pd.sub[sample(1:nrow(pd.sub), nrow(pd.sub), replace=TRUE),] 
  p_resample = princomp(data_resample,cor = T) 
  return(sum(p_resample$sdev[1:3]^2)/sum(p_resample$sdev^2))
  }

pco = data.frame(per=sapply(1:num_boot, bootstrap_it)) 
quantile(pco$per, probs = c(0.025,0.975))  # specify 95-th % Confidence Interval

##      2.5%     97.5% 
## 0.8134438 0.9035291

corpp = sum(pca1$sdev[1:3]^2)/sum(pca1$sdev^2)
require(ggplot2)
plot = ggplot(pco, aes(x=pco$per)) +
  geom_histogram() + geom_vline(xintercept=corpp, color='yellow')+ 
  labs(title = "Percent Var Explained by the first 3 PCs") +
  theme(plot.title = element_text(hjust = 0.5))+
  labs(x='perc of var')
show(plot)

## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.

Suppose we want to fit a linear model Top_of_SN_Voxel_Intensity_Ratio ~ Side_of_SN_Voxel_Intensity_Ratio + Part_IA. We can plot the 3D scatterplot and the (univariate) linear model.

library(scatterplot3d)

#Fit linear model
lm.fit <- lm(Top_of_SN_Voxel_Intensity_Ratio ~  Side_of_SN_Voxel_Intensity_Ratio + Part_IA, data = pd.sub)

#plot results
myPlot <- scatterplot3d(pd.sub$Side_of_SN_Voxel_Intensity_Ratio, pd.sub$Part_IA, pd.sub$Top_of_SN_Voxel_Intensity_Ratio)

# Get the ranges of the variable.names
summary(pd.sub$Side_of_SN_Voxel_Intensity_Ratio)

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##  0.9306  0.9958  1.1110  1.1065  1.1978  1.3811

summary(pd.sub$Part_IA)

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   0.000   0.000   1.000   1.242   2.000   6.000

summary(pd.sub$Top_of_SN_Voxel_Intensity_Ratio)

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   1.058   1.334   1.485   1.532   1.755   2.149

# Plot the linear model (line in 3D)
myCoef <- lm.fit$coefficients
plotX <- seq(0.93, 1.4,length.out = 100)
plotY <- seq(0,6,length.out = 100)
plotZ <- myCoef[1] + myCoef[2]*plotX + myCoef[3]*plotY # linear model
#Add the linear model to the 3D scatterplot
myPlot$points3d(plotX,plotY,plotZ, type = "l", lwd=2, col = "red")

However, we can also plot in 3D a bivariate 2D plane model (e.g., lm.fit', orpca1`) for the 3D scatter (Top_of_SN_Voxel_Intensity_Ratio, Side_of_SN_Voxel_Intensity_Ratio, Part_IA). Below is an example using the linear model.

myPlot <- scatterplot3d(pd.sub$Side_of_SN_Voxel_Intensity_Ratio, pd.sub$Part_IA, pd.sub$Top_of_SN_Voxel_Intensity_Ratio)

# Static Plot
myPlot$plane3d(lm.fit, lty.box = "solid")

# planes3d(a, b, c, d, alpha = 0.5)
# planes3d draws planes using the parametrization a*x + b*y + c*z + d = 0.
# Multiple planes may be specified by giving multiple values for the normal
# vector (a, b, c) and the offset parameter d

Next are examples of plotting the 3D scatter along with the 2D PCA model using either rgl or plot_ly.

pca1 <- prcomp(as.matrix(cbind(pd.sub$Side_of_SN_Voxel_Intensity_Ratio, pd.sub$Part_IA, pd.sub$Top_of_SN_Voxel_Intensity_Ratio)), center = T); summary(pca1)

## Importance of components:
##                           PC1     PC2     PC3
## Standard deviation     1.5218 0.27556 0.09958
## Proportion of Variance 0.9643 0.03162 0.00413
## Cumulative Proportion  0.9643 0.99587 1.00000

# Given two vectors PCA1 and PCA2, the cross product V = PCA1 x PCA2

# is orthogonal to both A and to B, and a normal vector to the 
# plane containing PCA1 and PCA2
# If PCA1 = (a,b,c) and PCA2 = (d, e, f), then the cross product is
# PCA1 x PCA2 =  (bf - ce, cd - af, ae - bd)
# PCA1 = pca1$rotation[,1] and PCAS2=pca1$rotation[,2]
# https://en.wikipedia.org/wiki/Cross_product#Names
#normVec = c(pca1$rotation[,1][2]*pca1$rotation[,2][3]-
#              pca1$rotation[,1][3]*pca1$rotation[,2][2],
#            pca1$rotation[,1][3]*pca1$rotation[,2][1]-
#              pca1$rotation[,1][1]*pca1$rotation[,2][3],
#            pca1$rotation[,1][1]*pca1$rotation[,2][2]-
#              pca1$rotation[,1][2]*pca1$rotation[,2][1]
#            )
normVec = c(pca1$rotation[2,1]*pca1$rotation[3,2]-
              pca1$rotation[3,1]*pca1$rotation[2,2],
            pca1$rotation[3,1]*pca1$rotation[1,2]-
              pca1$rotation[1,1]*pca1$rotation[3,2],
            pca1$rotation[1,1]*pca1$rotation[2,2]-
              pca1$rotation[2,1]*pca1$rotation[1,2]
            )

# Interactive RGL 3D plot with PCA Plane
library(rgl) 

# Compute the 3D point representing the gravitational balance
dMean <- apply( cbind(pd.sub$Side_of_SN_Voxel_Intensity_Ratio, pd.sub$Top_of_SN_Voxel_Intensity_Ratio, pd.sub$Part_IA), 2, mean)
# then the offset plane parameter is (d):
d <- as.numeric((-1)*normVec %*% dMean)  # force the plane to go through the mean

# Plot the PCA Plane
plot3d(pd.sub$Side_of_SN_Voxel_Intensity_Ratio, pd.sub$Part_IA, pd.sub$Top_of_SN_Voxel_Intensity_Ratio, type = "s", col = "red", size = 1)
planes3d(normVec[1], normVec[2], normVec[3], d, alpha = 0.5)

Another alternative is to use plot_ly for the interactive 3D visualization. First, we demonstrate displaying the lm() derived plane modeling superimposed on the 3D scatterplot (using pd.sub$Side_of_SN_Voxel_Intensity_Ratio, pd.sub$Top_of_SN_Voxel_Intensity_Ratio, and pd.sub$Part_IA).

# Define the 3D features
x <- pd.sub$Side_of_SN_Voxel_Intensity_Ratio
y <- pd.sub$Top_of_SN_Voxel_Intensity_Ratio
z <- pd.sub$Part_IA
myDF <- data.frame(x, y, z)

### Fit a (bivariate-predictor) linear regression model
lm.fit <- lm(z ~ x+y)
coef.lm.fit <- coef(lm.fit)

### Reparameterize the 2D (x,y) grid, and define the corresponding model values z on the grid
x.seq <- seq(min(x),max(x),length.out=100)
y.seq <- seq(min(y),max(y),length.out=100)
z.seq <- function(x,y) coef.lm.fit[1]+coef.lm.fit[2]*x+coef.lm.fit[3]*y
# define the values of z = z(x.seq, y.seq), as a Matrix of dimension c(dim(x.seq), dim(y.seq))
z <- t(outer(x.seq, y.seq, z.seq))

# First draw the 2D plane embedded in 3D, and then add points with "add_trace"
library(plotly)
myPlotly <- plot_ly(x=~x.seq, y=~y.seq, z=~z,
    colors = c("blue", "red"),type="surface", opacity=0.7) %>%
  
  add_trace(data=myDF, x=x, y=y, z=pd.sub$Part_IA, mode="markers", 
            type="scatter3d", marker = list(color="green", opacity=0.9, 
                                            symbol=105)) %>%
  
  layout(scene = list(
    aspectmode = "manual", aspectratio = list(x=1, y=1, z=1),
    xaxis = list(title = "Side_of_SN_Voxel_Intensity_Ratio"),
    yaxis = list(title = "Top_of_SN_Voxel_Intensity_Ratio"),
    zaxis = list(title = "Part_IA"))
    )
# print(myPlotly)
myPlotly

Second, we show the PCA-derived 2D plane model superimposed on the 3D scatterplot.

# define the original 3D coordinates
x <- pd.sub$Side_of_SN_Voxel_Intensity_Ratio
y <- pd.sub$Top_of_SN_Voxel_Intensity_Ratio
z <- pd.sub$Part_IA
myDF <- data.frame(x, y, z)

### Fit (compute) the 2D PCA space (dimensionaltiy reduction)
pca1 <- prcomp(as.matrix(cbind(pd.sub$Side_of_SN_Voxel_Intensity_Ratio, pd.sub$Top_of_SN_Voxel_Intensity_Ratio, pd.sub$Part_IA)), center = T); summary(pca1)

## Importance of components:
##                           PC1     PC2     PC3
## Standard deviation     1.5218 0.27556 0.09958
## Proportion of Variance 0.9643 0.03162 0.00413
## Cumulative Proportion  0.9643 0.99587 1.00000

# Compute the Normal to the 2D PC plane
normVec = c(pca1$rotation[2,1]*pca1$rotation[3,2]-
              pca1$rotation[3,1]*pca1$rotation[2,2],
            pca1$rotation[3,1]*pca1$rotation[1,2]-
              pca1$rotation[1,1]*pca1$rotation[3,2],
            pca1$rotation[1,1]*pca1$rotation[2,2]-
              pca1$rotation[2,1]*pca1$rotation[1,2]
            )
# Compute the 3D point of gravitational balance (Plane has to go through it)
dMean <- apply( cbind(pd.sub$Side_of_SN_Voxel_Intensity_Ratio, pd.sub$Top_of_SN_Voxel_Intensity_Ratio, pd.sub$Part_IA), 2, mean)

d <- as.numeric((-1)*normVec %*% dMean)  # force the plane to go through the mean

# Reparameterize the 2D (x,y) grid, and define the corresponding model values z on the grid. Recall z=-(d + ax+by)/c, where normVec=(a,b,c)
x.seq <- seq(min(x),max(x),length.out=100)
y.seq <- seq(min(y),max(y),length.out=100)
z.seq <- function(x,y) -(d + normVec[1]*x + normVec[2]*y)/normVec[3]
# define the values of z = z(x.seq, y.seq), as a Matrix of dimension c(dim(x.seq), dim(y.seq))
#z <- t(outer(x.seq, y.seq, z.seq))/10; range(z)  # we need to check this 10 correction, to ensure the range of z is apporproate!!!
z <- t(outer(x.seq, y.seq, z.seq)); range(z)

## [1] -31.19195  29.19934

# Draw the 2D plane embedded in 3D, and then add points with "add_trace"
library(plotly)
myPlotly <- plot_ly(x=~x.seq, y=~y.seq, z=~z,
    colors = c("blue", "red"),type="surface", opacity=0.7) %>%
  
  add_trace(data=myDF, x=x, y=y, z=pd.sub$Part_IA, mode="markers", 
            type="scatter3d", marker = list(color="green", opacity=0.9, 
                                            symbol=105)) %>%
  
  layout(scene = list(
    aspectmode = "manual", aspectratio = list(x=1, y=1, z=1),
    xaxis = list(title = "Side_of_SN_Voxel_Intensity_Ratio", range = c(min(x),max(x))),
    yaxis = list(title = "Top_of_SN_Voxel_Intensity_Ratio", range = c(min(y),max(y))),
    zaxis = list(title = "Part_IA"))
    )
# print(myPlotly)
myPlotly

As stated in the summary table, classical PCA assumes that the bivariate relations are linear in nature. The non-linear PCA is a generalization that allows us to incorporate nominal and ordinal variables, as well as to handle and identify nonlinear relationships between variables in the dataset. See Chapter 2 of this textbook Nonparametric inference in nonlinear principal components analysis: exploration and beyond.

Non-linear PCA assigns values to the categories representing the numeric variables, which maximize the association (e.g., correlation) between the quantified variables (i.e., optimal scaling to quantify the variables according to their analysis levels). The Bioconductor’s pcaMethods package provides the funcitonality for non-linear PCA.

6 Independent component analysis (ICA)

ICA aims to find basis vectors representing independent components of the original data. For example, this may be achieved by maximizing the norm of the \(4^{th}\) order normalized kurtosis, which iteratively projects the signal on a new basis vector, computes the objective function (e.g., the norm of the kurtosis) of the result, slightly adjusts the basis vector (e.g., by gradient ascent), and recomputes the kurtosis again. At the end, this iterative process generates a basis vector corresponding to the highest (residual) kurtosis representing the next independent component.

The process of Independent Component Analysis is to maximize the statistical independence of the estimated components. Assume that each variable \(X_i\) is generated by a sum of n independent components. \[X_i=a_{i, 1}s_1+...+a_{i, n}s_n\] Here, \(X_i\) is generated by \(s_1, ..., s_n\) and \(a_{i,1}, ... a_{i,n}\) are the corresponding weights. Finally, we rewrite \(X\) as \[X=As,\] where \(X=(X_1, ..., X_n)^T\), \(A=(a_1, ..., a_n)^T\), \(a_i=(a_{i,1}, ..., a_{i,n})\) and \(s=(s_1, ..., s_n)^T\). Note that \(s\) is obtained by maximizing the independence of the components. This procedure is done by maximizing some independence objective function.

ICA does not assume that all of its components (\(s_i\)) are Gaussian and independent of each other.

We will utilize the fastICA function in R.

fastICA(X, n.comp, alg.typ, fun, rownorm, maxit, tol)

• X: data matrix
• n.comp:number of components,
• alg.type: components extracted simultaneously(alg.typ == "parallel") or one at a time(alg.typ == "deflation")
• fun: functional form of F to approximate to neg-entropy,
• rownorm: whether rows of the data matrix X should be standardized beforehand
• maxit: maximum number of iterations
• tol: a positive scalar giving the tolerance at which the un-mixing matrix is considered to have converged.

Now we can create a correlated matrix \(X\).

S <- matrix(runif(10000), 5000, 2)
S[1:10, ]

##             [,1]       [,2]
##  [1,] 0.19032887 0.92326457
##  [2,] 0.64582044 0.36716717
##  [3,] 0.09673674 0.51115358
##  [4,] 0.24813471 0.03997883
##  [5,] 0.51746238 0.03503276
##  [6,] 0.94568595 0.86846372
##  [7,] 0.29500222 0.76227787
##  [8,] 0.93488888 0.97061365
##  [9,] 0.89622932 0.62092241
## [10,] 0.33758057 0.84543862

A <- matrix(c(1, 1, -1, 3), 2, 2, byrow = TRUE)
X <- S %*% A # In R,  "*" and "%*%" indicate "scalar" and matrix multiplication, respectively!
cor(X)

##            [,1]       [,2]
## [1,]  1.0000000 -0.4563297
## [2,] -0.4563297  1.0000000

The correlation between two variables is -0.4. Then we can start to fit the ICA model.

# install.packages("fastICA")
library(fastICA)
a <- fastICA(X, 2, alg.typ = "parallel", fun = "logcosh", alpha = 1, 
                 method = "C", row.norm = FALSE, maxit = 200, 
                 tol = 0.0001)

To visualize how correlated the pre-processed data is and how independent our \(S\) is we can draw the following two plots.

par(mfrow = c(1, 2))
plot(a$X, main = "Pre-processed data")
plot(a$S, main = "ICA components")

Finally we can confirm that the correlation of two components is nearly 0.

cor(a$S)

##               [,1]          [,2]
## [1,]  1.000000e+00 -7.677818e-16
## [2,] -7.677818e-16  1.000000e+00

Let’s look at a more interesting example, based on the pd.sub dataset. It has 6 variables and the correlation is relatively high. After fitting the ICA model. The components are nearly independent.

cor(pd.sub)

##                                  Top_of_SN_Voxel_Intensity_Ratio
## Top_of_SN_Voxel_Intensity_Ratio                       1.00000000
## Side_of_SN_Voxel_Intensity_Ratio                      0.54747225
## Part_IA                                              -0.10144191
## Part_IB                                              -0.26966299
## Part_II                                              -0.04358545
## Part_III                                             -0.33921790
##                                  Side_of_SN_Voxel_Intensity_Ratio
## Top_of_SN_Voxel_Intensity_Ratio                         0.5474722
## Side_of_SN_Voxel_Intensity_Ratio                        1.0000000
## Part_IA                                                -0.2157587
## Part_IB                                                -0.4438992
## Part_II                                                -0.3766388
## Part_III                                               -0.5226128
##                                     Part_IA    Part_IB     Part_II
## Top_of_SN_Voxel_Intensity_Ratio  -0.1014419 -0.2696630 -0.04358545
## Side_of_SN_Voxel_Intensity_Ratio -0.2157587 -0.4438992 -0.37663875
## Part_IA                           1.0000000  0.4913169  0.50378157
## Part_IB                           0.4913169  1.0000000  0.57987562
## Part_II                           0.5037816  0.5798756  1.00000000
## Part_III                          0.5845831  0.6735584  0.63901337
##                                    Part_III
## Top_of_SN_Voxel_Intensity_Ratio  -0.3392179
## Side_of_SN_Voxel_Intensity_Ratio -0.5226128
## Part_IA                           0.5845831
## Part_IB                           0.6735584
## Part_II                           0.6390134
## Part_III                          1.0000000

a1<-fastICA(pd.sub, 2, alg.typ = "parallel", fun = "logcosh", alpha = 1, 
                 method = "C", row.norm = FALSE, maxit = 200, 
                 tol = 0.0001)
par(mfrow = c(1, 2))
cor(a1$X)

##             [,1]       [,2]       [,3]       [,4]        [,5]       [,6]
## [1,]  1.00000000  0.5474722 -0.1014419 -0.2696630 -0.04358545 -0.3392179
## [2,]  0.54747225  1.0000000 -0.2157587 -0.4438992 -0.37663875 -0.5226128
## [3,] -0.10144191 -0.2157587  1.0000000  0.4913169  0.50378157  0.5845831
## [4,] -0.26966299 -0.4438992  0.4913169  1.0000000  0.57987562  0.6735584
## [5,] -0.04358545 -0.3766388  0.5037816  0.5798756  1.00000000  0.6390134
## [6,] -0.33921790 -0.5226128  0.5845831  0.6735584  0.63901337  1.0000000

cor(a1$S)

##              [,1]         [,2]
## [1,] 1.000000e+00 1.088497e-15
## [2,] 1.088497e-15 1.000000e+00

Notice that we only have 2 components instead of 6 variables. We successfully reduced the dimension of the data.

7 Factor analysis (FA)

Similar to ICA and PCA, FA tries to find special principal components in data. As a generalization of PCA, FA requires that the number of components is smaller than the original number of variables (or columns of the data matrix). FA optimization relies on iterative perturbations with full-dimensional Gaussian noise and maximum-likelihood estimation where every observation in the data represents a sample point in a higher dimensional space. Whereas PCA assumes the noise is spherical, Factor Analysis allows the noise to have an arbitrary diagonal covariance matrix and estimates the subspace as well as the noise covariance matrix.

Under FA, the centered data can be expressed in the following from:

\[x_i-\mu_i=l_{i, 1}F_1+...+l_{i, k}F_k+\epsilon_i=LF+\epsilon_i,\]

where \(i\in {1, ..., p}\), \(j \in{1, ..., k}\), \(k<p\) and \(\epsilon_i\) are independently distributed error terms with zero mean and finite variance.

Let’s try FA in R using the function factanal(). According to the previous PCA, our pd.sub dataset can explain 95% of variance using only the first two principal components. This suggest that we might need 2 factors in FA. We can double check that by examining the scree plot.

## Report For a nScree Class 
## 
## Details: components 
## 
##   Eigenvalues Prop Cumu Par.Analysis Pred.eig     OC Acc.factor     AF
## 1           3    1    1            1        1                NA (< AF)
## 2           1    0    1            1        1 (< OC)          1       
## 3           1    0    1            1        0                 1       
## 4           0    0    1            1        0                 0       
## 5           0    0    1            1       NA                 0       
## 6           0    0    1            0       NA                NA       
## 
## 
##  Number of factors retained by index 
## 
##   noc naf nparallel nkaiser
## 1   2   1         2       2

Note that 3 out of 4 Cattell’s Scree test rules summary suggest we should use 2 factors. Thus, in the function factanal() we can specify factors=2. In addition, we can use varimax rotation of the factor axes maximizing the variance of the squared loadings of factors (columns) on the original variables (rows), which effectively differentiates the original variables by the extracted factors. Oblique promax and Procrustes rotation (projecting the loadings to a target matrix with a simple structure) are two alternative and commonly used matrix rotations that may be specified.

fit<-factanal(pd.sub, factors=2, rotation="varimax")
# fit<-factanal(pd.sub, factors=2, rotation="promax") # the most popular oblique rotation; And fitting a simple structure
fit

## 
## Call:
## factanal(x = pd.sub, factors = 2, rotation = "varimax")
## 
## Uniquenesses:
##  Top_of_SN_Voxel_Intensity_Ratio Side_of_SN_Voxel_Intensity_Ratio 
##                            0.018                            0.534 
##                          Part_IA                          Part_IB 
##                            0.571                            0.410 
##                          Part_II                         Part_III 
##                            0.392                            0.218 
## 
## Loadings:
##                                  Factor1 Factor2
## Top_of_SN_Voxel_Intensity_Ratio           0.991 
## Side_of_SN_Voxel_Intensity_Ratio -0.417   0.540 
## Part_IA                           0.650         
## Part_IB                           0.726  -0.251 
## Part_II                           0.779         
## Part_III                          0.825  -0.318 
## 
##                Factor1 Factor2
## SS loadings      2.412   1.445
## Proportion Var   0.402   0.241
## Cumulative Var   0.402   0.643
## 
## Test of the hypothesis that 2 factors are sufficient.
## The chi square statistic is 1.35 on 4 degrees of freedom.
## The p-value is 0.854

Here the p-value 0.854 is very large, suggesting that we failed to reject the null-hypothesis that 2 factors are sufficient. We can also visualize the loadings for all the variables.

load <- fit$loadings
plot(load, type="n") # set up plot 
text(load, labels=colnames(pd.sub), cex=.7) # add variable names

This plot displays factors 1 and 2 on the x-axis and y-axis, respectively.

8 Singular Value Decomposition (SVD)

SVD is a factorization of a real or complex matrix. If we have a data matrix \(X\) with \(n\) observation and \(p\) variables it can be factorized into the following form: \[X=U D V^T,\] where \(U\) is a \(n \times p\) unitary matrix that \(U^TU=I\), \(D\) is a \(p \times p\) diagonal matrix, and \(V^T\) is a \(p \times p\) unitary matrix, which is the conjugate transpose of the \(n\times n\) unitary matrix, \(V\). Thus, we have \(V^TV=I\).

SVD is closely linked to PCA (when correlation matrix is used for calculation). \(U\) represents the left singular vectors, \(D\) the singular values, \(U%*%D\) yields the PCA scores, and \(V\) the right singular vectors - PCA loadings.

Using the pd.sub dataset, we can compare the outputs from the svd() function and the princomp() function (another R function for PCA). Prior to the SVD, we need to scale the data matrix.

#SVD output
df<-nrow(pd.sub)-1
zvars<-scale(pd.sub)
z.svd<-svd(zvars)
z.svd$d/sqrt(df)

## [1] 1.7878123 1.1053808 0.7550519 0.6475685 0.5688743 0.5184536

z.svd$v

##            [,1]       [,2]        [,3]        [,4]       [,5]        [,6]
## [1,]  0.2555204 0.71258155 -0.37323594  0.10487773 -0.4773992  0.22073161
## [2,]  0.3855208 0.47213743  0.35665523 -0.43312945  0.5581867  0.04564469
## [3,] -0.3825033 0.37288211  0.70992668  0.31993403 -0.2379855 -0.22728693
## [4,] -0.4597352 0.09803466 -0.11166513 -0.79389290 -0.2915570 -0.22647775
## [5,] -0.4251107 0.34167997 -0.46424927  0.26165346  0.5341197 -0.36505061
## [6,] -0.4976933 0.06258370  0.03872473 -0.01769966  0.1832789  0.84438182

#PCA output
pca2<-princomp(pd.sub, cor=T)
pca2

## Call:
## princomp(x = pd.sub, cor = T)
## 
## Standard deviations:
##    Comp.1    Comp.2    Comp.3    Comp.4    Comp.5    Comp.6 
## 1.7878123 1.1053808 0.7550519 0.6475685 0.5688743 0.5184536 
## 
##  6  variables and  33 observations.

loadings(pca2)

## 
## Loadings:
##                                  Comp.1 Comp.2 Comp.3 Comp.4 Comp.5 Comp.6
## Top_of_SN_Voxel_Intensity_Ratio   0.256  0.713  0.373  0.105  0.477  0.221
## Side_of_SN_Voxel_Intensity_Ratio  0.386  0.472 -0.357 -0.433 -0.558       
## Part_IA                          -0.383  0.373 -0.710  0.320  0.238 -0.227
## Part_IB                          -0.460         0.112 -0.794  0.292 -0.226
## Part_II                          -0.425  0.342  0.464  0.262 -0.534 -0.365
## Part_III                         -0.498                      -0.183  0.844
## 
##                Comp.1 Comp.2 Comp.3 Comp.4 Comp.5 Comp.6
## SS loadings     1.000  1.000  1.000  1.000  1.000  1.000
## Proportion Var  0.167  0.167  0.167  0.167  0.167  0.167
## Cumulative Var  0.167  0.333  0.500  0.667  0.833  1.000

When correlation matrix is used for calculation (cor=T), the \(V\) matrix of SVD contains the corresponding PCA loadings.

8.1 SVD Summary

Intuitively, the SVD approach \(X= UD V^T\) represents a composition of the (centered!) data into 3 geometrical transformations: a rotation or reflection (\(U\)), a scaling (\(D\)), and a rotation or reflection (\(V\)). Here we assume that the data \(X\) stores samples/cases in rows and variables/features in columns. If these are reversed, then the interpretations of the \(U\) and \(V\) matrices reverse as well.

• The columns of \(V\) represent the directions of the principal axes, the columns of \(UD\) are the principal components, and the singular values in \(D\) are related to the eigenvalues of data variance-covariance matrix (\(\Sigma\)) via \(\lambda_i = \frac{d_i^2}{n-1}\), where the eigenvalues \(\lambda_i\) capture the magnitude of the data variance in the respective PCs.
• The standardized scores are given by columns of \(\sqrt{n-1}U\), the corresponding loadings are given by columns of \(\frac{1}{n-1}VD\). However, these “loadings” are not the principal directions. The requirement for \(X\) to be centered is needed to ensure that the covariance matrix \(Cov(X) = \frac{1}{n-1}X^TX\).
• Alternatively, to perform PCA on the correlation matrix (instead of the covariance matrix), the columns of \(X\) need to be scaled (centered and standardized).
• Reduce the data dimensionality from \(p\) to \(k<p\), we multiply the first \(k\) columns of \(U\) by the \(k\times k\) upper-left corner of the matrix \(D\) to get an \(n\times k\) matrix \(U_kD_k\) containing the first \(k\) PCs.
• Multiplying the first \(k\) PCs by their corresponding principal directions \(V_k^T\) reconstructs the original data from the first \(k\) PCs, \(X_k=U_k D_kV_k^T\), with the lowest possible reconstruction error.
• Typically we have more subjects/cases (\(n\)) than variables/features (\(p<n\)). As \(U_{n\times n}\) and \(V_{p\times p}\), the last \(n-p>0\) columns of \(U\) may be trivial (zeros). It’s customary to drop the zero columns of \(U\) for \(n>>p\) to avid dealing with unnecessarily large (trivial) matrices.

9 t-SNE (t-distributed Stochastic Neighbor Embedding)

The t-SNE technique represents a recent machine learning strategy for nonlinear dimensionality reduction that is useful for embedding (e.g., scatter-plotting) of high-dimensional data into lower-dimensional (1D, 2D, 3D) spaces. For each object (point in the high-dimensional space), the method models similar objects using nearby and dissimilar objects using remote distant objects. The two steps in t-SNE include (1) construction of a probability distribution over pairs of the original high-dimensional objects where similar objects have a high probability of being paired and correspondingly, dissimilar objects have a small probability of being selected; and (2) defining a similar probability distribution over the points in the derived low-dimensional embedding minimizing the Kullback-Leibler divergence between the high- and low-dimensional distributions relative to the locations of the objects in the embedding map. Either Euclidean or non-Euclidean distance measures between objects may be used as similarity metrics.

9.1 t-SNE Formulation

Suppose we have high dimensional data (\(N\)D): \(x_1, x_2,..., x_N\). In step 1, for each pair (\(x_i, x_j\)), t-SNE estimates the probabilities \(p_{i,j}\) that are proportional to their corresponding similarities, \(p_{j | i}\):

\[p_{j | i} = \frac{\exp\left (\frac{-||x_i - x_j||^2}{2\sigma_i^2} \right )}{\sum_{k \neq i} \exp\left (\frac{-||x_i - x_k||^2}{2\sigma_i^2} \right )}.\]

The similarity between \(x_j\) and \(x_i\) may be thought of as the conditional probability, \(p_{j | i}\). That is, assuming \(N\)D Gaussian distributions centered at each point \(x_i\), neighbors are selected based on a probability distribution (proportion of their probability density), which represents the chance that \(x_i\) may select \(x_j\) as its neighbor, \(p_{i,j} = \frac{p_{j | i} + p_{i |j}}{2N}\).

The perplexity (\(perp\)) of a discrete probability distribution, \(p\), is defined as an exponential function of the entropy, \(H(p)\), over all discrete events: \(perp(x)=2^{H(p)}=2^{-\sum _{x}p(x)\log_{2}p(x)}\). t-SNE performs a binary search for the value \(\sigma_i\) that produces a predefined value \(perp\). The simple interpretation of the perplexity at a data point \(x_i\), \(2^{H(p_i)}\), is as a smooth measure of the effective number of points in the \(x_i\) neighborhood. The performance of t-SNE may vary with the perplexity value, which is typically specified by the user, e.g., between \(5\leq perp\leq 50\).

Then, the precision (variance, \(\sigma_i\)) of the local Gaussian kernels may be chosen to ensure that the perplexity of the conditional distribution equals a specified perplexity. This allows adapting the kernel bandwidth to the sample data density – smaller \(\sigma_i\) values are fitted in denser areas of the sample data space, and correspondingly, larger \(\sigma_i\) are fitted in sparser areas. A particular value of \(\sigma_i\) yields a probability distribution, \(p_i\), over all of other data points, which has an increasing entropy as \(\sigma_i\) increases.

t-SNE learns a mapping \(f: \{x_1, x_2, ..., x_N\} \longrightarrow \{y_1, y_2, ..., y_d\}\), where \(x_i\in R^N\) and \(y_i \in R^d\) (\(N\gg d\)) that resembles closely the original similarities, \(p_{i,j}\) and represents the derived similarities, \(q_{i,j}\) between pairs of embedded points \(y_i,y_j\), defined by:

\[q_{i,j} = \frac{(1 + ||y_i - y_j||^2)^{-1}}{\sum_{k \neq i} (1 + ||y_i - y_k||^2)^{-1}}.\]

The t-distributed reference in t-SNE refers to the heavy-tailed Student-t distribution (\(t_{df=1}\)) which councides with Cauchy distribution, \(f(z)=\frac{1}{1+z^2}\). It is used to model and measure similarities between closer points in the embedded low-dimensional space, as well as dissimilarities of objects that map far apart in the embedded space.

The rationale for using Student t distribution for mapping the points is based on the fact that the volume of an \(N\)D ball of radius \(r\), \(B^N\), is proportional to \(r^N\). Specifically, \(V_N(r) = \frac{\pi^\frac{N}{2}}{\Gamma\left(\frac{N}{2} + 1\right)}r^N\), where \(\Gamma()\) is the Euler’s gamma function, which is an extension of the factorial function to non-integer arguments. For large \(N\), when we select uniformly random points inside \(B^N\), most points will be expected to be close to the ball surface (boundary), \(S^{N-1}\), and few will be expected near the \(B^N\) center, as half the volume of \(B^N\) is included in the hyper-area inside \(B^N\) and outside a ball of radius \(r_1=\frac{1}{\sqrt[N]{2}}\times r \sim r\). You can try this with \(N=2\), \(\{x\in R^2 |\ ||x||\leq r\}\), representing a disk in a 2D plane.

When reducing the dimensionality of a dataset, if we used the Gaussian distribution for the mapping embedding into the lower dimensional space, there will be a distortion of the distribution of the distances between neighboring objects. This is simply because the distribution of the distances is much different between the original (high-dimensional) and a the map-transformed low-dimensional spaces. t-SNE tries to (approximately) preserve the distances in the two spaces to avoid imbalances that may lead to biases due to excessive attraction-repulsion forces. Using Student t distribution \(df=1\) (aka Cauchy distribution) for mapping the points preserves (to some extent) the distance similarity distribution, because of the heavier tails of \(t\) compared to the Gaussian distribution. For a given similarity between a pair of data points, the two corresponding map points will need to be much further apart in order for their similarity to match the data similarity.

A minimization process with respect to the objects \(y_i\) using gradient descent of a (non-symmetric) objective function, Kullback-Leibler divergence between the distributions \(Q\) and \(P\) , is used to determine the object locations \(y_i\) in the map, i.e.,

\[KL(P || Q) = \sum_{i \neq j} p_{i,j} \log \frac{p_{i,j}}{q_{i,j}}.\]

The minimization of the KL objective function by gradient descent may be analytically represented by:

\[\frac{\partial {KL(P||Q)}}{\partial {y_i}}= \sum_{j}{(p_{i,j}-q_{i,j})f(|x_i-x_j|) u_{i,j}},\] where \(f(z)=\frac{z}{1+z^2}\) and \(u_{i,j}\) is a unit vector from \(y_j\) to \(y_i\). This gradient represents the aggregate sum of all spring forces applied to map point \(x_i\).

This optimization leads to an embedding mapping that “preserves” the object (data point) similarities of the original high-dimensional inputs into the lower dimensional space. Note that the data similarity matrix (\(p_{i,j}\)) is fixed, whereas its counterpart, the map similarity matrix (\(q_{i,j}\)) depends on the embedding map. Of course, we want these two distance matrices to be as close as possible, implying that similar data points in the original space yield similar map-points in the reduced dimension.

9.2 t-SNE Example: Hand-written Digit Recognition

Later, in Chapter 10 and Chapter 22, we will present the Optical Character Recognition (OCR) and analysis of hand-written notes (unstructured text).

Below, we show a simple example of generating a 2D embedding of the hand-written digits dataset using t-SNE.

# install.packages("tsne"); library (tsne)
# install.packages("Rtsne")
library(Rtsne)

# Download the hand-written digits data
pathToZip <- tempfile() 
download.file("http://www.socr.umich.edu/people/dinov/2017/Spring/DSPA_HS650/data/DigitRecognizer_TrainingData.zip", pathToZip) 
train <- read.csv(unzip(pathToZip)) 
dim(train)  

## [1] 42000   785

unlink(pathToZip)   

# identify the label-nomenclature - digits 0, 1, 2, ..., 9 - and map to diff colors
colMap <- function(x){
  cols <- rainbow(length(x))[order(order(x))] # reindexing by ranking the observed values
}

# Note on "order(order())": set.seed(123); x <- sample(10)
# This experiment shows that order(order()) = rank()
# set.seed(12345); data <- sample(6); data
# order(data); order(order(data)); rank(data)
# Ordering acts as its own inverse and returns a sequence starting with the index of the
# smallest data value (1). Nested odd "order" applications yield the same vector outputs.
# Order outputs an index vector useful for sorting the original data vector. 
# The location of the smallest value is in the first position of the order-output.
# The index of the second smallest data value is next, etc.
# The last order output item represents the index of the largest data value.
# Double-order application yields an indexing vector where the first element is the index 
# of the smallest first-order-index, etc., which corresponds to the data vector's rank.

train.labels<-train$label
train$label<-as.factor(train$label)
train.labels.colors <- colMap(train.labels)
names(train.labels.colors) <- train$label # unique(train$label)

# May need to check and increase the RAM allocation
memory.limit() 

## [1] 32647

memory.limit(50000)

## [1] 50000

# Remove the labels (column 1) and Scale the image intensities to [0; 1]
train  <- data.matrix(train[, -1]); dim(train)

## [1] 42000   784

train <- t(train/255)

# Visualize some of the images
library("imager")
# first convert the CSV data (one row per image, 42,000 rows)
array_3D <- array(train[ , ], c(28, 28, 42000))
mat_2D <- matrix(array_3D[,,1], nrow = 28, ncol = 28)
plot(as.cimg(mat_2D))

N <- 42000
img_3D <- as.cimg(array_3D[,,], 28, 28, N)

# plot the k-th image (1<=k<=N)
k <- 5; plot(img_3D, k)

k <- 6; plot(img_3D, k)

k <- 7; plot(img_3D, k)

pretitle <- function(index) bquote(bold("Image: "~.(index)))
#layout(t(1:2))
op <- par(mfrow = c(2,2), oma = c(5,4,0,0) + 0.1, mar = c(0,0,1,1) + 0.1)

for (k in 1:4) {
  plot(img_3D, k, xlim = c(0,28), ylim = c(28,0), axes=F, ann=T, main=pretitle(k))
}

# Run the t-SNE, tracking the execution time (artificially reducing the sample-size to get reasonable calculation time)
execTime_tSNE <- system.time(tsne_digits <- Rtsne(t(train)[1:10000 , ], dims = 2, perplexity=30, verbose=TRUE, max_iter = 500)); execTime_tSNE

## Read the 10000 x 50 data matrix successfully!
## Using no_dims = 2, perplexity = 30.000000, and theta = 0.500000
## Computing input similarities...
## Normalizing input...
## Building tree...
##  - point 0 of 10000
## Done in 6.55 seconds (sparsity = 0.012259)!
## Learning embedding...
## Iteration 50: error is 97.574505 (50 iterations in 3.24 seconds)
## Iteration 100: error is 90.668312 (50 iterations in 3.28 seconds)
## Iteration 150: error is 86.528260 (50 iterations in 3.10 seconds)
## Iteration 200: error is 86.151440 (50 iterations in 3.48 seconds)
## Iteration 250: error is 86.099838 (50 iterations in 3.47 seconds)
## Iteration 300: error is 3.142056 (50 iterations in 2.92 seconds)
## Iteration 350: error is 2.745428 (50 iterations in 2.67 seconds)
## Iteration 400: error is 2.526139 (50 iterations in 2.68 seconds)
## Iteration 450: error is 2.384572 (50 iterations in 2.81 seconds)
## Iteration 500: error is 2.283056 (50 iterations in 2.65 seconds)
## Fitting performed in 30.28 seconds.

##    user  system elapsed 
##   52.44    0.89   53.67

# Full dataset(42K * 1K) execution may take over 5-mins
# execTime_tSNE <- system.time(tsne_digits <- Rtsne(train[ , ], dims = 2, perplexity=30, verbose=TRUE, max_iter = 500)); execTime_tSNE

# Plot the result 2D map embedding of the data with digits or solid discs
par(mfrow=c(1,1)) # reset the plot canvas

# Plot only first 1,000 cases (to avoid clutter)
plot(tsne_digits$Y[1:1000, ], t='n', main="t-SNE") # don't plot the points to avoid clutter
text(tsne_digits$Y[1:1000, ], labels=names(train.labels.colors)[1:1000], col=train.labels.colors[1:1000])

# plot all cases as solid discs with colors corresponding to each of the 10 numbers
plot(tsne_digits$Y, main="t-SNE Clusters", col=train.labels.colors, pch = 19)
legend("topright", unique(names(train.labels.colors)), fill=unique(train.labels.colors), bg='gray90', cex=0.5)

The hands-on interactive SOCR t-SNE Dimensionaltiy Reduction Activity provides an interactive demonstration of t-SNE utilizing TensorBoard and the UK Biobank data.

10 Dimensionality Reduction Case Study (Parkinson’s Disease)

10.1 Step 1: Collecting Data

The data we will be using in this case study is the Clinical, Genetic and Imaging Data for Parkinson’s Disease in the SOCR website. A detailed data explanation is on the following link PD data. Let’s import the data into R.

# Loading required package: xml2
wiki_url <- read_html("http://wiki.socr.umich.edu/index.php/SOCR_Data_PD_BiomedBigMetadata")
html_nodes(wiki_url, "#content")

## {xml_nodeset (1)}
## [1] <div id="content" class="mw-body-primary" role="main">\n\t<a id="top ...

pd_data <- html_table(html_nodes(wiki_url, "table")[[1]])
head(pd_data); summary(pd_data)

##   Cases L_caudate_ComputeArea L_caudate_Volume R_caudate_ComputeArea
## 1     2                   597              767                   855
## 2     2                   597              767                   855
## 3     2                   597              767                   855
## 4     2                   597              767                   855
## 5     3                   604              873                   935
## 6     3                   604              873                   935
##   R_caudate_Volume L_putamen_ComputeArea L_putamen_Volume
## 1              968                   842             1357
## 2              968                   842             1357
## 3              968                   842             1357
## 4              968                   842             1357
## 5             1043                   892             1366
## 6             1043                   892             1366
##   R_putamen_ComputeArea R_putamen_Volume L_hippocampus_ComputeArea
## 1                  1285             3052                      1306
## 2                  1285             3052                      1306
## 3                  1285             3052                      1306
## 4                  1285             3052                      1306
## 5                  1305             2920                      1292
## 6                  1305             2920                      1292
##   L_hippocampus_Volume R_hippocampus_ComputeArea R_hippocampus_Volume
## 1                 3238                      1513                 3759
## 2                 3238                      1513                 3759
## 3                 3238                      1513                 3759
## 4                 3238                      1513                 3759
## 5                 3079                      1516                 3827
## 6                 3079                      1516                 3827
##   cerebellum_ComputeArea cerebellum_Volume L_lingual_gyrus_ComputeArea
## 1                  16845             13949                        3268
## 2                  16845             13949                        3268
## 3                  16845             13949                        3268
## 4                  16845             13949                        3268
## 5                  16698             14076                        3243
## 6                  16698             14076                        3243
##   L_lingual_gyrus_Volume R_lingual_gyrus_ComputeArea
## 1                  11130                        3294
## 2                  11130                        3294
## 3                  11130                        3294
## 4                  11130                        3294
## 5                  11033                        3190
## 6                  11033                        3190
##   R_lingual_gyrus_Volume L_fusiform_gyrus_ComputeArea
## 1                  12221                         3625
## 2                  12221                         3625
## 3                  12221                         3625
## 4                  12221                         3625
## 5                  12187                         3631
## 6                  12187                         3631
##   L_fusiform_gyrus_Volume R_fusiform_gyrus_ComputeArea
## 1                   11087                         3232
## 2                   11087                         3232
## 3                   11087                         3232
## 4                   11087                         3232
## 5                   11116                         3302
## 6                   11116                         3302
##   R_fusiform_gyrus_Volume Sex Weight Age Dx chr12_rs34637584_GT
## 1                   10122   1     84  67 PD                   1
## 2                   10122   1     84  67 PD                   1
## 3                   10122   1     84  67 PD                   1
## 4                   10122   1     84  67 PD                   1
## 5                   10162   0     97  39 PD                   1
## 6                   10162   0     97  39 PD                   1
##   chr17_rs11868035_GT UPDRS_part_I UPDRS_part_II UPDRS_part_III Time
## 1                   0            1            12              1    0
## 2                   0            1            12              1    6
## 3                   0            1            12              1   12
## 4                   0            1            12              1   18
## 5                   1            0            19             22    0
## 6                   1            0            19             22    6

##      Cases       L_caudate_ComputeArea L_caudate_Volume
##  Min.   :  2.0   Min.   :525.0         Min.   :719.0   
##  1st Qu.:158.0   1st Qu.:582.0         1st Qu.:784.0   
##  Median :363.5   Median :600.0         Median :800.0   
##  Mean   :346.1   Mean   :600.4         Mean   :800.3   
##  3rd Qu.:504.0   3rd Qu.:619.0         3rd Qu.:819.0   
##  Max.   :692.0   Max.   :667.0         Max.   :890.0   
##  R_caudate_ComputeArea R_caudate_Volume L_putamen_ComputeArea
##  Min.   :795.0         Min.   : 916     Min.   : 815.0       
##  1st Qu.:875.0         1st Qu.: 979     1st Qu.: 879.0       
##  Median :897.0         Median : 998     Median : 897.5       
##  Mean   :894.5         Mean   :1001     Mean   : 898.9       
##  3rd Qu.:916.0         3rd Qu.:1022     3rd Qu.: 919.0       
##  Max.   :977.0         Max.   :1094     Max.   :1003.0       
##  L_putamen_Volume R_putamen_ComputeArea R_putamen_Volume
##  Min.   :1298     Min.   :1198          Min.   :2846    
##  1st Qu.:1376     1st Qu.:1276          1st Qu.:2959    
##  Median :1400     Median :1302          Median :3000    
##  Mean   :1400     Mean   :1300          Mean   :3000    
##  3rd Qu.:1427     3rd Qu.:1321          3rd Qu.:3039    
##  Max.   :1507     Max.   :1392          Max.   :3148    
##  L_hippocampus_ComputeArea L_hippocampus_Volume R_hippocampus_ComputeArea
##  Min.   :1203              Min.   :3036         Min.   :1414             
##  1st Qu.:1277              1st Qu.:3165         1st Qu.:1479             
##  Median :1300              Median :3200         Median :1504             
##  Mean   :1302              Mean   :3198         Mean   :1504             
##  3rd Qu.:1325              3rd Qu.:3228         3rd Qu.:1529             
##  Max.   :1422              Max.   :3381         Max.   :1602             
##  R_hippocampus_Volume cerebellum_ComputeArea cerebellum_Volume
##  Min.   :3634         Min.   :16378          Min.   :13680    
##  1st Qu.:3761         1st Qu.:16617          1st Qu.:13933    
##  Median :3802         Median :16699          Median :13996    
##  Mean   :3799         Mean   :16700          Mean   :14002    
##  3rd Qu.:3833         3rd Qu.:16784          3rd Qu.:14077    
##  Max.   :4013         Max.   :17096          Max.   :14370    
##  L_lingual_gyrus_ComputeArea L_lingual_gyrus_Volume
##  Min.   :3136                Min.   :10709         
##  1st Qu.:3262                1st Qu.:10943         
##  Median :3299                Median :11007         
##  Mean   :3300                Mean   :11010         
##  3rd Qu.:3333                3rd Qu.:11080         
##  Max.   :3469                Max.   :11488         
##  R_lingual_gyrus_ComputeArea R_lingual_gyrus_Volume
##  Min.   :3135                Min.   :11679         
##  1st Qu.:3258                1st Qu.:11935         
##  Median :3294                Median :12001         
##  Mean   :3296                Mean   :12008         
##  3rd Qu.:3338                3rd Qu.:12079         
##  Max.   :3490                Max.   :12324         
##  L_fusiform_gyrus_ComputeArea L_fusiform_gyrus_Volume
##  Min.   :3446                 Min.   :10682          
##  1st Qu.:3554                 1st Qu.:10947          
##  Median :3594                 Median :11016          
##  Mean   :3598                 Mean   :11011          
##  3rd Qu.:3637                 3rd Qu.:11087          
##  Max.   :3763                 Max.   :11394          
##  R_fusiform_gyrus_ComputeArea R_fusiform_gyrus_Volume      Sex        
##  Min.   :3094                 Min.   : 9736           Min.   :0.0000  
##  1st Qu.:3260                 1st Qu.: 9928           1st Qu.:0.0000  
##  Median :3296                 Median : 9994           Median :1.0000  
##  Mean   :3299                 Mean   : 9996           Mean   :0.5851  
##  3rd Qu.:3332                 3rd Qu.:10058           3rd Qu.:1.0000  
##  Max.   :3443                 Max.   :10235           Max.   :1.0000  
##      Weight            Age             Dx            chr12_rs34637584_GT
##  Min.   : 51.00   Min.   :31.00   Length:1128        Min.   :0.000      
##  1st Qu.: 71.00   1st Qu.:54.00   Class :character   1st Qu.:0.000      
##  Median : 78.50   Median :61.00   Mode  :character   Median :1.000      
##  Mean   : 78.45   Mean   :60.64                      Mean   :0.539      
##  3rd Qu.: 84.00   3rd Qu.:68.00                      3rd Qu.:1.000      
##  Max.   :109.00   Max.   :87.00                      Max.   :1.000      
##  chr17_rs11868035_GT  UPDRS_part_I   UPDRS_part_II    UPDRS_part_III 
##  Min.   :0.0000      Min.   :0.000   Min.   : 1.000   Min.   : 1.00  
##  1st Qu.:0.0000      1st Qu.:0.000   1st Qu.: 5.000   1st Qu.: 6.00  
##  Median :0.0000      Median :1.000   Median : 9.000   Median :13.00  
##  Mean   :0.4184      Mean   :0.773   Mean   : 8.879   Mean   :13.02  
##  3rd Qu.:1.0000      3rd Qu.:1.000   3rd Qu.:13.000   3rd Qu.:18.00  
##  Max.   :1.0000      Max.   :2.000   Max.   :20.000   Max.   :30.00  
##       Time     
##  Min.   : 0.0  
##  1st Qu.: 4.5  
##  Median : 9.0  
##  Mean   : 9.0  
##  3rd Qu.:13.5  
##  Max.   :18.0

10.2 Step 2: Exploring and preparing the data

To make sure that the data is ready for further modeling, we need to fix a few things. Firstly, the Dx variable or diagnosis is a factor. We need to change it to a numeric variable. Second, we don’t need the patient ID and time variable in the dimension reduction procedures.

pd_data$Dx <- gsub("PD", 1, pd_data$Dx)
pd_data$Dx <- gsub("HC", 0, pd_data$Dx)
pd_data$Dx <- gsub("SWEDD", 0, pd_data$Dx)
pd_data$Dx <- as.numeric(pd_data$Dx)
attach(pd_data)
pd_data<-pd_data[, -c(1, 33)]

10.3 PCA

Now we start the process of fitting a PCA model. Here we will use the princomp() function and use the correlation rather than the covariance matrix for calculation.

pca.model <- princomp(pd_data, cor=TRUE)
summary(pca.model) # pc loadings (i.e., eigenvector columns)

## Importance of components:
##                            Comp.1     Comp.2     Comp.3    Comp.4
## Standard deviation     1.39495952 1.28668145 1.28111293 1.2061402
## Proportion of Variance 0.06277136 0.05340481 0.05294356 0.0469282
## Cumulative Proportion  0.06277136 0.11617617 0.16911973 0.2160479
##                            Comp.5     Comp.6   Comp.7     Comp.8    Comp.9
## Standard deviation     1.18527282 1.15961464 1.135510 1.10882348 1.0761943
## Proportion of Variance 0.04531844 0.04337762 0.041593 0.03966095 0.0373611
## Cumulative Proportion  0.26136637 0.30474399 0.346337 0.38599794 0.4233590
##                           Comp.10    Comp.11    Comp.12    Comp.13
## Standard deviation     1.06687730 1.05784209 1.04026215 1.03067437
## Proportion of Variance 0.03671701 0.03609774 0.03490791 0.03426741
## Cumulative Proportion  0.46007604 0.49617378 0.53108169 0.56534910
##                          Comp.14    Comp.15    Comp.16    Comp.17
## Standard deviation     1.0259684 0.99422375 0.97385632 0.96688855
## Proportion of Variance 0.0339552 0.03188648 0.03059342 0.03015721
## Cumulative Proportion  0.5993043 0.63119078 0.66178421 0.69194141
##                           Comp.18    Comp.19    Comp.20    Comp.21
## Standard deviation     0.92687735 0.92376374 0.89853718 0.88924412
## Proportion of Variance 0.02771296 0.02752708 0.02604416 0.02550823
## Cumulative Proportion  0.71965437 0.74718145 0.77322561 0.79873384
##                           Comp.22    Comp.23    Comp.24    Comp.25
## Standard deviation     0.87005195 0.86433816 0.84794183 0.82232529
## Proportion of Variance 0.02441905 0.02409937 0.02319372 0.02181351
## Cumulative Proportion  0.82315289 0.84725226 0.87044598 0.89225949
##                           Comp.26    Comp.27    Comp.28    Comp.29
## Standard deviation     0.80703739 0.78546699 0.77505522 0.76624322
## Proportion of Variance 0.02100998 0.01990188 0.01937776 0.01893963
## Cumulative Proportion  0.91326947 0.93317135 0.95254911 0.97148875
##                           Comp.30    Comp.31
## Standard deviation     0.68806884 0.64063259
## Proportion of Variance 0.01527222 0.01323904
## Cumulative Proportion  0.98676096 1.00000000

plot(pca.model)

biplot(pca.model)

fviz_pca_biplot(pca.model, axes = c(1, 2), geom = "point",
  col.ind = "black", col.var = "steelblue", label = "all",
  invisible = "none", repel = F, habillage = pd_data$Sex, 
  palette = NULL, addEllipses = TRUE, title = "PCA - Biplot")

Albeit the two cohorts (normal controls and patients) are slightly separated in the second principal direction, we can see in this real world example that PCs do not necessarily correspond to a definitive “elbow” plot suggesting an optimal number of components. In our PCA model, each PC explains about the same amount of variation. Thus, it is hard to tell how many PCs, or factors, we need to select. This would be an ad hoc decision in this case. We can understand this better after understanding the following FA model.

10.4 FA

Let’s set up an Cattel’s Scree test to determine the number of factors first.

ev <- eigen(cor(pd_data)) # get eigenvalues
ap <- parallel(subject=nrow(pd_data), var=ncol(pd_data), rep=100, cent=.05)
nS <- nScree(x=ev$values, aparallel=ap$eigen$qevpea)
summary(nS)

## Report For a nScree Class 
## 
## Details: components 
## 
##    Eigenvalues Prop Cumu Par.Analysis Pred.eig     OC Acc.factor     AF
## 1            2    0    0            1        2 (< OC)         NA (< AF)
## 2            2    0    0            1        2                 0       
## 3            2    0    0            1        1                 0       
## 4            1    0    0            1        1                 0       
## 5            1    0    0            1        1                 0       
## 6            1    0    0            1        1                 0       
## 7            1    0    0            1        1                 0       
## 8            1    0    0            1        1                 0       
## 9            1    0    0            1        1                 0       
## 10           1    0    0            1        1                 0       
## 11           1    0    0            1        1                 0       
## 12           1    0    1            1        1                 0       
## 13           1    0    1            1        1                 0       
## 14           1    0    1            1        1                 0       
## 15           1    0    1            1        1                 0       
## 16           1    0    1            1        1                 0       
## 17           1    0    1            1        1                 0       
## 18           1    0    1            1        1                 0       
## 19           1    0    1            1        1                 0       
## 20           1    0    1            1        1                 0       
## 21           1    0    1            1        1                 0       
## 22           1    0    1            1        1                 0       
## 23           1    0    1            1        1                 0       
## 24           1    0    1            1        1                 0       
## 25           1    0    1            1        1                 0       
## 26           1    0    1            1        1                 0       
## 27           1    0    1            1        1                 0       
## 28           1    0    1            1        1                 0       
## 29           1    0    1            1        1                 0       
## 30           0    0    1            1       NA                 0       
## 31           0    0    1            1       NA                NA       
## 
## 
##  Number of factors retained by index 
## 
##   noc naf nparallel nkaiser
## 1   1   1        14      14

Although the Cattel’s Scree test suggest that we should use 14 factors, the real fit shows 14 is not enough. Previous PCA results suggest we need around 20 PCs to obtain a cumulative variance of 0.6. After a few trials we find that 19 factors can pass the chi square test for sufficient number of factors at \(0.05\) level.

fa.model<-factanal(pd_data, 19, rotation="varimax")
fa.model

## 
## Call:
## factanal(x = pd_data, factors = 19, rotation = "varimax")
## 
## Uniquenesses:
##        L_caudate_ComputeArea             L_caudate_Volume 
##                        0.840                        0.005 
##        R_caudate_ComputeArea             R_caudate_Volume 
##                        0.868                        0.849 
##        L_putamen_ComputeArea             L_putamen_Volume 
##                        0.791                        0.702 
##        R_putamen_ComputeArea             R_putamen_Volume 
##                        0.615                        0.438 
##    L_hippocampus_ComputeArea         L_hippocampus_Volume 
##                        0.476                        0.777 
##    R_hippocampus_ComputeArea         R_hippocampus_Volume 
##                        0.798                        0.522 
##       cerebellum_ComputeArea            cerebellum_Volume 
##                        0.137                        0.504 
##  L_lingual_gyrus_ComputeArea       L_lingual_gyrus_Volume 
##                        0.780                        0.698 
##  R_lingual_gyrus_ComputeArea       R_lingual_gyrus_Volume 
##                        0.005                        0.005 
## L_fusiform_gyrus_ComputeArea      L_fusiform_gyrus_Volume 
##                        0.718                        0.559 
## R_fusiform_gyrus_ComputeArea      R_fusiform_gyrus_Volume 
##                        0.663                        0.261 
##                          Sex                       Weight 
##                        0.829                        0.005 
##                          Age                           Dx 
##                        0.005                        0.005 
##          chr12_rs34637584_GT          chr17_rs11868035_GT 
##                        0.638                        0.721 
##                 UPDRS_part_I                UPDRS_part_II 
##                        0.767                        0.826 
##               UPDRS_part_III 
##                        0.616 
## 
## Loadings:
##                              Factor1 Factor2 Factor3 Factor4 Factor5
## L_caudate_ComputeArea                                               
## L_caudate_Volume                                              0.980 
## R_caudate_ComputeArea                                               
## R_caudate_Volume                                                    
## L_putamen_ComputeArea                                               
## L_putamen_Volume                                                    
## R_putamen_ComputeArea                                               
## R_putamen_Volume                                                    
## L_hippocampus_ComputeArea                                           
## L_hippocampus_Volume                                                
## R_hippocampus_ComputeArea    -0.102                                 
## R_hippocampus_Volume                                                
## cerebellum_ComputeArea                                              
## cerebellum_Volume                                                   
## L_lingual_gyrus_ComputeArea   0.107                                 
## L_lingual_gyrus_Volume                                              
## R_lingual_gyrus_ComputeArea                           0.989         
## R_lingual_gyrus_Volume                0.983                         
## L_fusiform_gyrus_ComputeArea                                        
## L_fusiform_gyrus_Volume                                             
## R_fusiform_gyrus_ComputeArea                                        
## R_fusiform_gyrus_Volume                                             
## Sex                                          -0.111                 
## Weight                                        0.983                 
## Age                                                                 
## Dx                            0.965                                 
## chr12_rs34637584_GT                   0.124                         
## chr17_rs11868035_GT          -0.303                                 
## UPDRS_part_I                 -0.260                                 
## UPDRS_part_II                                                       
## UPDRS_part_III                0.332           0.104                 
##                              Factor6 Factor7 Factor8 Factor9 Factor10
## L_caudate_ComputeArea                -0.101                          
## L_caudate_Volume                                                     
## R_caudate_ComputeArea                                                
## R_caudate_Volume                     -0.103          -0.107  -0.182  
## L_putamen_ComputeArea                                 0.299  -0.147  
## L_putamen_Volume                                             -0.123  
## R_putamen_ComputeArea                                 0.147  -0.175  
## R_putamen_Volume                                      0.698          
## L_hippocampus_ComputeArea                                            
## L_hippocampus_Volume                                                 
## R_hippocampus_ComputeArea                                            
## R_hippocampus_Volume                                                 
## cerebellum_ComputeArea                0.920                          
## cerebellum_Volume                                             0.690  
## L_lingual_gyrus_ComputeArea   0.106                           0.143  
## L_lingual_gyrus_Volume                                               
## R_lingual_gyrus_ComputeArea                                          
## R_lingual_gyrus_Volume                                               
## L_fusiform_gyrus_ComputeArea                                         
## L_fusiform_gyrus_Volume                                              
## R_fusiform_gyrus_ComputeArea                                         
## R_fusiform_gyrus_Volume                       0.844                  
## Sex                                                                  
## Weight                                                               
## Age                           0.984                                  
## Dx                                                                   
## chr12_rs34637584_GT                          -0.195  -0.207   0.197  
## chr17_rs11868035_GT                                  -0.165          
## UPDRS_part_I                                         -0.209          
## UPDRS_part_II                                                        
## UPDRS_part_III                               -0.161                  
##                              Factor11 Factor12 Factor13 Factor14 Factor15
## L_caudate_ComputeArea                                             0.113  
## L_caudate_Volume                                                         
## R_caudate_ComputeArea                           0.174   -0.164           
## R_caudate_Volume              0.174                               0.125  
## L_putamen_ComputeArea                          -0.165                    
## L_putamen_Volume                                0.128   -0.149           
## R_putamen_ComputeArea         0.225             0.260                    
## R_putamen_Volume                                                         
## L_hippocampus_ComputeArea              0.708                             
## L_hippocampus_Volume                                                     
## R_hippocampus_ComputeArea                                         0.331  
## R_hippocampus_Volume          0.652            -0.114                    
## cerebellum_ComputeArea                                                   
## cerebellum_Volume                                                        
## L_lingual_gyrus_ComputeArea           -0.126             0.136    0.137  
## L_lingual_gyrus_Volume                                                   
## R_lingual_gyrus_ComputeArea                                              
## R_lingual_gyrus_Volume                                                   
## L_fusiform_gyrus_ComputeArea                                      0.493  
## L_fusiform_gyrus_Volume                                  0.646           
## R_fusiform_gyrus_ComputeArea  0.121            -0.544                    
## R_fusiform_gyrus_Volume                                                  
## Sex                                                                      
## Weight                                                                   
## Age                                                                      
## Dx                                                                       
## chr12_rs34637584_GT                             0.227                    
## chr17_rs11868035_GT                             0.168   -0.113    0.206  
## UPDRS_part_I                  0.212    0.122                     -0.123  
## UPDRS_part_II                                                            
## UPDRS_part_III                0.104                     -0.121   -0.282  
##                              Factor16 Factor17 Factor18 Factor19
## L_caudate_ComputeArea        -0.119   -0.165    0.237           
## L_caudate_Volume                                                
## R_caudate_ComputeArea                                   -0.112  
## R_caudate_Volume                       0.120             0.113  
## L_putamen_ComputeArea                                    0.164  
## L_putamen_Volume              0.382   -0.187   -0.131           
## R_putamen_ComputeArea                 -0.218   -0.109    0.341  
## R_putamen_Volume             -0.128             0.110           
## L_hippocampus_ComputeArea                                       
## L_hippocampus_Volume                  -0.106            -0.435  
## R_hippocampus_ComputeArea     0.181                             
## R_hippocampus_Volume                                            
## cerebellum_ComputeArea                                          
## cerebellum_Volume                                               
## L_lingual_gyrus_ComputeArea   0.256                      0.140  
## L_lingual_gyrus_Volume                          0.536           
## R_lingual_gyrus_ComputeArea                                     
## R_lingual_gyrus_Volume                                          
## L_fusiform_gyrus_ComputeArea -0.113                             
## L_fusiform_gyrus_Volume                                         
## R_fusiform_gyrus_ComputeArea                                    
## R_fusiform_gyrus_Volume                                         
## Sex                          -0.352   -0.111                    
## Weight                        0.106                             
## Age                                                             
## Dx                                     0.210                    
## chr12_rs34637584_GT          -0.289    0.186   -0.152           
## chr17_rs11868035_GT                                     -0.175  
## UPDRS_part_I                                    0.127           
## UPDRS_part_II                          0.378                    
## UPDRS_part_III                         0.311                    
## 
##                Factor1 Factor2 Factor3 Factor4 Factor5 Factor6 Factor7
## SS loadings      1.282   1.029   1.026   1.019   1.013   1.011   0.921
## Proportion Var   0.041   0.033   0.033   0.033   0.033   0.033   0.030
## Cumulative Var   0.041   0.075   0.108   0.140   0.173   0.206   0.235
##                Factor8 Factor9 Factor10 Factor11 Factor12 Factor13
## SS loadings      0.838   0.782    0.687    0.647    0.615    0.587
## Proportion Var   0.027   0.025    0.022    0.021    0.020    0.019
## Cumulative Var   0.263   0.288    0.310    0.331    0.351    0.370
##                Factor14 Factor15 Factor16 Factor17 Factor18 Factor19
## SS loadings       0.569    0.566    0.547    0.507    0.475    0.456
## Proportion Var    0.018    0.018    0.018    0.016    0.015    0.015
## Cumulative Var    0.388    0.406    0.424    0.440    0.455    0.470
## 
## Test of the hypothesis that 19 factors are sufficient.
## The chi square statistic is 54.51 on 47 degrees of freedom.
## The p-value is 0.211

This data matrix has relatively low correlation. Thus, it is not suitable for ICA.

cor(pd_data)[1:10, 1:10]

##                           L_caudate_ComputeArea L_caudate_Volume
## L_caudate_ComputeArea               1.000000000       0.05794916
## L_caudate_Volume                    0.057949162       1.00000000
## R_caudate_ComputeArea              -0.060576361       0.01076372
## R_caudate_Volume                    0.043994457       0.07245568
## L_putamen_ComputeArea               0.009640983      -0.06632813
## L_putamen_Volume                   -0.064299184      -0.11131525
## R_putamen_ComputeArea               0.040808105       0.04504867
## R_putamen_Volume                    0.058552841      -0.11830387
## L_hippocampus_ComputeArea          -0.037932760      -0.04443615
## L_hippocampus_Volume               -0.042033469      -0.04680825
##                           R_caudate_ComputeArea R_caudate_Volume
## L_caudate_ComputeArea              -0.060576361      0.043994457
## L_caudate_Volume                    0.010763720      0.072455677
## R_caudate_ComputeArea               1.000000000      0.057441889
## R_caudate_Volume                    0.057441889      1.000000000
## L_putamen_ComputeArea              -0.015959528     -0.017003442
## L_putamen_Volume                    0.063279351      0.021962691
## R_putamen_ComputeArea               0.078643479      0.054287467
## R_putamen_Volume                    0.007022844     -0.094336376
## L_hippocampus_ComputeArea           0.051359613      0.006123355
## L_hippocampus_Volume                0.085788328     -0.077913614
##                           L_putamen_ComputeArea L_putamen_Volume
## L_caudate_ComputeArea               0.009640983      -0.06429918
## L_caudate_Volume                   -0.066328127      -0.11131525
## R_caudate_ComputeArea              -0.015959528       0.06327935
## R_caudate_Volume                   -0.017003442       0.02196269
## L_putamen_ComputeArea               1.000000000       0.02228947
## L_putamen_Volume                    0.022289469       1.00000000
## R_putamen_ComputeArea               0.090496109       0.09093926
## R_putamen_Volume                    0.176353726      -0.05768765
## L_hippocampus_ComputeArea           0.094604791       0.02530330
## L_hippocampus_Volume               -0.064425367       0.04041557
##                           R_putamen_ComputeArea R_putamen_Volume
## L_caudate_ComputeArea                0.04080810      0.058552841
## L_caudate_Volume                     0.04504867     -0.118303868
## R_caudate_ComputeArea                0.07864348      0.007022844
## R_caudate_Volume                     0.05428747     -0.094336376
## L_putamen_ComputeArea                0.09049611      0.176353726
## L_putamen_Volume                     0.09093926     -0.057687648
## R_putamen_ComputeArea                1.00000000      0.052245264
## R_putamen_Volume                     0.05224526      1.000000000
## L_hippocampus_ComputeArea           -0.05508472      0.131800075
## L_hippocampus_Volume                -0.08866344     -0.001133570
##                           L_hippocampus_ComputeArea L_hippocampus_Volume
## L_caudate_ComputeArea                  -0.037932760          -0.04203347
## L_caudate_Volume                       -0.044436146          -0.04680825
## R_caudate_ComputeArea                   0.051359613           0.08578833
## R_caudate_Volume                        0.006123355          -0.07791361
## L_putamen_ComputeArea                   0.094604791          -0.06442537
## L_putamen_Volume                        0.025303302           0.04041557
## R_putamen_ComputeArea                  -0.055084723          -0.08866344
## R_putamen_Volume                        0.131800075          -0.00113357
## L_hippocampus_ComputeArea               1.000000000          -0.02633816
## L_hippocampus_Volume                   -0.026338163           1.00000000

10.5 t-SNE

Finally, let’s try the t-Distributed Stochastic Neighbor Embedding method on the PD data.

# install.packages("Rtsne")
library(Rtsne)

# If working with post-processed PD data above: remove duplicates (after stripping time)
# pd_data <- unique(pd_data[,])

# If wqorking with raw PD data: reload it
pd_data <- html_table(html_nodes(wiki_url, "table")[[1]])

# Run the t-SNE, tracking the execution time (artificially reducing the sample-size to get reasonable calculation time)
execTime_tSNE <- system.time(tsne_digits <- Rtsne(pd_data, dims = 2, perplexity=30, verbose=TRUE, max_iter = 1000)); execTime_tSNE

## Read the 1128 x 35 data matrix successfully!
## Using no_dims = 2, perplexity = 30.000000, and theta = 0.500000
## Computing input similarities...
## Normalizing input...
## Building tree...
##  - point 0 of 1128
## Done in 0.18 seconds (sparsity = 0.111894)!
## Learning embedding...
## Iteration 50: error is 71.955842 (50 iterations in 0.25 seconds)
## Iteration 100: error is 69.349938 (50 iterations in 0.25 seconds)
## Iteration 150: error is 69.325872 (50 iterations in 0.22 seconds)
## Iteration 200: error is 69.326291 (50 iterations in 0.22 seconds)
## Iteration 250: error is 69.326107 (50 iterations in 0.20 seconds)
## Iteration 300: error is 1.306181 (50 iterations in 0.31 seconds)
## Iteration 350: error is 0.949250 (50 iterations in 0.47 seconds)
## Iteration 400: error is 0.835112 (50 iterations in 0.53 seconds)
## Iteration 450: error is 0.792189 (50 iterations in 0.58 seconds)
## Iteration 500: error is 0.771158 (50 iterations in 0.63 seconds)
## Iteration 550: error is 0.761531 (50 iterations in 0.70 seconds)
## Iteration 600: error is 0.753071 (50 iterations in 0.78 seconds)
## Iteration 650: error is 0.748645 (50 iterations in 0.85 seconds)
## Iteration 700: error is 0.745904 (50 iterations in 0.92 seconds)
## Iteration 750: error is 0.741102 (50 iterations in 0.98 seconds)
## Iteration 800: error is 0.738204 (50 iterations in 1.04 seconds)
## Iteration 850: error is 0.735856 (50 iterations in 1.07 seconds)
## Iteration 900: error is 0.733867 (50 iterations in 1.14 seconds)
## Iteration 950: error is 0.733031 (50 iterations in 1.17 seconds)
## Iteration 1000: error is 0.732359 (50 iterations in 1.24 seconds)
## Fitting performed in 13.55 seconds.

##    user  system elapsed 
##   13.72    0.42   14.16

# Plot the result 2D map embedding of the data
# table(pd_data$Sex)
# plot(tsne_digits$Y, main="t-SNE Clusters", col=rainbow(length(unique(pd_data$Sex))), pch = 1)
#legend("topright", c("Male", "Female"), fill=rainbow(length(unique(pd_data$Sex))), bg='gray90', cex=0.5)

table(pd_data$Dx)

## 
##    HC    PD SWEDD 
##   400   400   328

# Either use the DX label column to set the colors col = as.factor(pd_data$Dx) 
#plot(tsne_digits$Y, main="t-SNE Clusters", col=as.factor(pd_data$Dx), pch = 15)
#legend("topright", c("HC", "PD", "SWEDD"), fill=unique(as.factor(pd_data$Dx)), bg='gray90', cex=0.5)


# Or to set the colors explicitly
CharToColor = function(input_char){ 
    mapping = c("HC"="blue", "PD"="red", "SWEDD"="yellow")
    mapping[input_char]
}
pd_data$Dx.col = sapply(pd_data$Dx, CharToColor)

plot(tsne_digits$Y, main="t-SNE Clusters", col=pd_data$Dx.col, pch = 15)
legend("topright", c("HC", "PD", "SWEDD"), fill=unique(pd_data$Dx.col), bg='gray90', cex=0.5)

The results of the PCA, ICA, FA, and t-SNE methods on the PD data imply that the data is complex and intrinsically high-dimensional, which prevents explicit embeddings into a low-dimensional (e.g., 2D) space. More advanced methods to interrogate this dataset will be demonstrated later. The SOCR publications site provides additional examples of Parkinson’s disease studies.

SOCR Resource Visitor number